JEE Advanced 2019 · Paper 2

Let P₁ = I = [1 0 0; 0 1 0; 0 0 1], P₂ = [1 0 0; 0 0 1; 0 1 0], P₃ = [0 1 0; 1 0 0; 0 0 1], P₄ = [0 1 0; 0 0 1; 1 0 0], P₅ = [0 0 1; 1 0 0; 0 1 0], P₆ = [0 0 1; 0 1 0; 1 0 0] and X = Σₖ₌₁⁶ Pₖ [2 1 3; 1 0 2; 3 2 1] Pₖᵀ, where Pₖᵀ denotes the transpose of the matrix Pₖ. Then which of the following options is/are correct?

1. AIf X[1;1;1] = α[1;1;1], then α = 30
2. BX is a symmetric matrix
3. CThe sum of diagonal entries of X is 18
4. DX − 30I is an invertible matrix

For non-negative integers n, let f(n) = (Σₖ₌₀ⁿ sin((k + 1)/(n + 2) π) sin((k + 2)/(n + 2) π))/(Σₖ₌₀ⁿ sin²((k + 1)/(n + 2) π)). Assuming cos⁻¹x takes values in [0, π], which of the following options is/are correct?

1. Af(4) = √3/2
2. Blimₙ→∞ f(n) = 1/2
3. CIf α = tan(cos⁻¹ f(6)), then α² + 2α − 1 = 0
4. Dsin(7 cos⁻¹ f(5)) = 0

Let f: ℝ → ℝ be a function. We say that f has PROPERTY 1 if limₕ→₀ (f(h) − f(0))/√|h| exists and is finite, and PROPERTY 2 if limₕ→₀ (f(h) − f(0))/h² exists and is finite. Then which of the following options is/are correct?

1. Af(x) = |x| has PROPERTY 1
2. Bf(x) = x²/³ has PROPERTY 1
3. Cf(x) = x|x| has PROPERTY 2
4. Df(x) = sin x has PROPERTY 2

Let f(x) = sin πx/x², x > 0. Let x₁ < x₂ < x₃ < ··· < xₙ < ··· be all the points of local maximum of f and y₁ < y₂ < y₃ < ··· < yₙ < ··· be all the points of local minimum of f. Then which of the following options is/are correct?

1. Ax₁ < y₁
2. Bxₙ₊₁ − xₙ > 2 for every n
3. Cxₙ ∈ (2n, 2n + 1/2) for every n
4. D|xₙ − yₙ| > 1 for every n

Let f: ℝ → ℝ be given by f(x) = (x − 1)(x − 2)(x − 5). Define F(x) = ∫₀ˣ f(t) dt, x > 0. Then which of the following options is/are correct?

1. AF has a local minimum at x = 1
2. BF has a local maximum at x = 2
3. CF has two local maxima and one local minimum in (0, ∞)
4. DF(x) ≠ 0 for all x ∈ (0, 5)

Three lines L₁: r⃗ = λî, λ ∈ ℝ, L₂: r⃗ = k̂ + μĵ, μ ∈ ℝ and L₃: r⃗ = î + ĵ + νk̂, ν ∈ ℝ are given. For which point(s) Q on L₂ can we find a point P on L₁ and a point R on L₃ so that P, Q and R are collinear?

1. Ak̂ − 1/2 ĵ
2. Bk̂
3. Ck̂ + 1/2 ĵ
4. Dk̂ + ĵ